Class field theory is based on certain duality properties of the 1-dimensional scheme \(X=\operatorname{Spec} \mathcal {O}_K\) . Let \(\mathbb {G}_{m, X}\)  be the étale sheaf on \(X\) , which associates to an étale cover \(\varphi :\operatorname{Spec} B\rightarrow X\) , the group of multiplicative units \(\mathbb {G}_{m, X}(\varphi )=B^{\times }\) . Given a constructible sheaf \(\mathfrak {M}\)  on \(X\)  (see [1]} for the definition), set \(\mathfrak {M}^\vee :=\underline{\operatorname{Hom}}(\mathfrak {M}, \mathbb {G}_{m, X})\)  (where \(\underline{\operatorname{Hom}}(\cdot , \cdot )\)  is standard notation for the sheaf of homomorphisms, see [2]} or loc. cit.). We denote by \(\hat{H}^i(X, \mathfrak {M})\)  the modified étale cohomology groups of \(\mathfrak {M}\) . The Artin-Verdier duality theorem (see [1]}) states that there is a canonical isomorphism \(\hat{H}^3(X, \mathbb {G}_{m, X})\simeq \mathbb {Q}/\mathbb {Z}\)  and a nondegenerate pairing of abelian groups
\(\hat{H}^i(X, \mathfrak {M}^\vee )\times \operatorname{Ext}_X^{3-i}(\mathfrak {M}, \mathbb {G}_{m, X})\rightarrow \hat{H}^3(X, \mathbb {G}_{m, X})\xrightarrow{} \mathbb {Q}/\mathbb {Z}.\) 
